Design and extrude an elleptical dome with a hole knowing a few points only

Hello Experts,
I am a newbie here and I would like to ask you an advice about the approach I should use.

I have a frame (Frame A, black plastic on an image below) and am generating another frame (Frame B), which should fit perfectly inside my existing Frame A.
I have issues with creating a body/extruding a figure which is best described as ‘the top of an elleptical dome’. Please see the following image of the frame A (existing one):

Everything is perfect and I am close. Now I reach the closest part (the elleptical dome) and I even have my main points arranged:
NOTE (not obvious): the distance between points is different on BOTH X and Z, not only X or Z.

QUESTION 1: how can I create a body from what I have? Basically I need an elleptical dome, and then subtract from it (the latter is easy).
QUESTION 2: is there any other (Better) way of creating the figure I have?
I am not asking to model for me, I am asking for the approach you might suggest, because at this moment I am out of the options.

Explanation for 1: I used Sweep tool, but it generates a body with many artifacts: there is a margin on the bottom and there is an artifact in the corner. Once I start moving them, the whole body deformes and is not usable anymore. I can have corner substracted, but I can not get rid of the margin (again, once I move the new body for at least 1 mm, it deforms, and my calculated points are not valid anymore)

Explanation for 2: I was thinking of generating a separate dome by drawing an ellips on XY, an arc on XZ and using Revolve on an ellips and then Loft it, and then subtract it in the middle. The main problem here, I can not calculate the proper dimensions for the arc, not even close. Once I move my figure, it never fits all the points I need.

I would appreciate the answer. Again, not asking for a model, but for the approach you might suggest. I am learning and your advice might help a lot.

And the result is:

Okay so I was able to achieve the result by using the Loft tool. Unfortunately, at least for me, Loft only worked with parallel lines/faces, therefore I had issues with it when I tried to loft with curves.
For the future generations, this is how I was able to do it.

If I may add my 2 cents here. You seem to be hung up on using Loft. Your basic frame is quite geometric in nature. Here’s a simple way to create the wedge shape. The vertical sketch plane is used to demonstrate two examples. The upper triangle sketch is used to subtract from the vertically extruded ‘U’ shaped body. The lower trapezoid sketch is used with the Intersect boolean. And, for the elliptical dome, I’d Revolve the half ellipse sketch to 180°.